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Real analysis relative to a finite measure space

In the traditional foundations of probability theory, one selects a probability space , and makes a distinction between deterministic mathematical objects, which do not depend on the sampled state , and… 9,801 more words

Expository

Algebraic probability spaces

As laid out in the foundational work of Kolmogorov, a classical probability space (or probability space for short) is a triplet , where is a set, is a -algebra of subsets of , and is a countably additive probability measure on . 2,721 more words

Expository

Lebesgue measure as the invariant factor of Loeb measure

There are a number of ways to construct the real numbers , for instance

Expository

An abstract ergodic theorem, and the Mackey-Zimmer theorem

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if is a unitary operator on a Hilbert space , and is a vector in that Hilbert space, then one has… 2,315 more words

Expository

When is correlation transitive?

Given two unit vectors in a real inner product space, one can define the correlation between these vectors to be their inner product , or in more geometric terms, the cosine of the angle subtended by and . 1,196 more words

Expository